Actuarial Mathematics for Life Contingent Risks

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How can actuaries equip themselves for the products and risk structures of the future? Using the powerful framework of multiple state models, three leaders in actuarial science give a modern perspective on life contingencies, and develop and demonstrate a theory that can be adapted to changing products and technologies. The book begins traditionally, covering actuarial models and theory, and emphasizing practical applications using computational techniques. The authors then develop a more contemporary outlook, introducing multiple state models, emerging cash flows and embedded options. Using spreadsheet-style software, the book presents large-scale, realistic examples. Over 150 exercises and solutions teach skills in simulation and projection through computational practice. Balancing rigor with intuition, and emphasizing applications, this text is ideal for university courses, but also for individuals preparing for professional actuarial exams and qualified actuaries wishing to freshen up their skills.

Preface

p. xiv

Introduction to life insurance

p. 1

Summary

p. 1

Background

p. 1

Life insurance and annuity contracts

p. 3

Introduction

p. 3

Traditional insurance contracts

p. 4

Modern insurance contracts

p. 6

Distribution methods

p. 8

Underwriting

p. 8

Premiums

p. 10

Life annuities

p. 11

Other insurance contracts

p. 12

Pension benefits

p. 12

Defined benefit and defined contribution pensions

p. 13

Defined benefit pension design

p. 13

Mutual and proprietary insurers

p. 14

Typical problems

p. 14

Notes and further reading

p. 15

Exercises

p. 15

Survival models

p. 17

Summary

p. 17

The future lifetime random variable

p. 17

The force of mortality

p. 21

Actuarial notation

p. 26

Mean and standard deviation of Tx

p. 29

Curtate future lifetime

p. 32

Kx and ex

p. 32

The complete and curtate expected future lifetimes, ex and ex

p. 34

Notes and further reading

p. 35

Exercises

p. 36

Life tables and selection

p. 41

Summary

p. 41

Life tables

p. 41

Fractional age assumptions

p. 44

Uniform distribution of deaths

p. 44

Constant force of mortality

p. 48

National life tables

p. 49

Survival models for life insurance policyholders

p. 52

Life insurance underwriting

p. 54

Select and ultimate survival models

p. 56

Notation and formulae for select survival models

p. 58

Select life tables

p. 59

Notes and further reading

p. 67

Exercises

p. 67

Insurance benefits

p. 73

Summary

p. 73

Introduction

p. 73

Assumptions

p. 74

Valuation of insurance benefits

p. 75

Whole life insurance: the continuous case, &Abar;x

p. 75

Whole life insurance: the annual case, Ax

p. 78

Whole life insurance: the 1 /mthly case, A(m)x

p. 79

Recursions

p. 81

Term insurance

p. 86

Pure endowment

p. 88

Endowment insurance

p. 89

Deferred insurance benefits

p. 91

Relating &Abar;x, Ax and A(m)x

p. 93

Using the uniform distribution of deaths assumption

p. 93

Using the claims acceleration approach

p. 95

Variable insurance benefits

p. 96

Functions for select lives

p. 101

Notes and further reading

p. 101

Exercises

p. 102

Annuities

p. 107

Summary

p. 107

Introduction

p. 107

Review of annuities-certain

p. 108

Annual life annuities

p. 108

Whole life annuity-due

p. 109

Term annuity-due

p. 112

Whole life immediate annuity

p. 113

Term immediate annuity

p. 114

Annuities payable continuously

p. 115

Whole life continuous annuity

p. 115

Term continuous annuity

p. 117

Annuities payable m times per year

p. 118

Introduction

p. 118

Life annuities payable m times a year

p. 119

Term annuities payable m times a year

p. 120

Comparison of annuities by payment frequency

p. 121

Deferred annuities

p. 123

Guaranteed annuities

p. 125

Increasing annuities

p. 127

Arithmetically increasing annuities

p. 127

Geometrically increasing annuities

p. 129

Evaluating annuity functions

p. 130

Recursions

p. 130

Applying the UDD assumption

p. 131

Woolhouse's formula

p. 132

Numerical illustrations

p. 135

Functions for select lives

p. 136

Notes and further reading

p. 137

Exercises

p. 137

Premium calculation

p. 142

Summary

p. 142

Preliminaries

p. 142

Assumptions

p. 143

The present value of future loss random variable

p. 145

The equivalence principle

p. 146

Net premiums

p. 146

Gross premium calculation

p. 150

Profit

p. 154

The portfolio percentile premium principle

p. 162

Extra risks

p. 165

Age rating

p. 165

Constant addition to żx

p. 165

Constant multiple of mortality rates

p. 167

Notes and further reading

p. 169

Exercises

p. 170

Policy values

p. 176

Summary

p. 176

Assumptions

p. 176

Policies with annual cash flows

p. 176

The future loss random variable

p. 176

Policy values for policies with annual cash flows

p. 182

Recursive formulae for policy values

p. 191

Annual profit

p. 196

Asset shares

p. 200

Policy values for policies with cash flows at discrete intervals other than annually